Electromagnetic waves exhibiting vortex properties from orbital angular momentum (OAM) modes that are inherently orthogonal show promising potential for a wide range of applications. However, classic OAM-wave generators in the radio-frequency band are generally limited by a narrow operation bandwidth, complicated feeding structures, or high losses. In this work, we propose a method for OAM-wave generation using the concept of spatial transformation at microwave frequencies. A reflection-type vortex-beam generator is designed and simulated. Then an all-dielectric proof-of-concept prototype for a +1 topological charge is physically implemented and the complete characterization of the device is reported. Both full-wave simulations and experimental measurements successfully validate the spiral-shaped phase fronts of the vortex wave. Furthermore, the all-dielectric implementation of the device enables operation over a broad frequency bandwidth. The proposed method provides an efficient approach to generate vortex waves carrying OAM modes and illustrates the practicality of using spatial transformations to achieve an alternative class of optical devices at microwave frequencies.